## Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system

Optics Express, Vol. 16, Issue 5, pp. 3397-3407 (2008)

http://dx.doi.org/10.1364/OE.16.003397

Acrobat PDF (1416 KB)

### Abstract

The Debye-Wolf electromagnetic diffraction integral is now routinely used to describe focusing by high numerical (NA) lenses. We obtain an eigenfunction expansion of the electric vector field in the focal region in terms of Bessel and generalized prolate spheroidal functions. Our representation has many optimal and desirable properties which offer considerable simplification to the evaluation and analysis of the Debye-Wolf integral. It is potentially also useful in implementing two-dimensional apodization techniques to synthesize electromagnetic field distributions in the focal region of a high NA lenses. Our work is applicable to many areas, such as optical microscopy, optical data storage and lithography.

© 2008 Optical Society of America

## 1. Introduction

4. G. N. Watson, *A Treatise on the Theory of Bessel Functions, 2 ^{nd} Ed.* (Cambridge University Press, 1952). [PubMed]

6. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. **69**, 575–578 (1979). [CrossRef]

*et al.*to express both the infocus and defocus terms by means of a series of analytic functions [8

8. P. Török, S.J. Hewlett, and P. Varga, “On the series expansion of high aperture, vectorial diffraction integrals,” J. Mod. Opt. **44**, 493–503 (1997). [CrossRef]

7. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with seidel aberrations” J. Mod. Opt. **40**, 2293–2310 (1993). [CrossRef]

9. S. S. Sherif and P. Török, “Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula,” J. Mod. Opt. **52**, 857–876 (2005). [CrossRef]

*I*integrals of Richards and Wolf. In reality these expansions do not reveal anything about the physical nature of the problem – they merely provide a simplified way to calculate the diffraction integrals when computational time is deemed to be of significance.

*et al.*obtained the field components in the focal region as a series using Nijboer-Zernike functions [10

10. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A **20**, 2281–2292 (2003). [CrossRef]

11. C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. **44**, 803–818 (1997). [CrossRef]

9. S. S. Sherif and P. Török, “Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula,” J. Mod. Opt. **52**, 857–876 (2005). [CrossRef]

13. M. R. Foreman, Department of Physics, Imperial College London, Prince Consort Rd. London SW7 2BZ, United Kingdom, S. S. Sherif, P.R.T. Munro, and P. Török are preparing a manuscript to be called “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region”

## 2. Focusing of high numerical aperture beams

*p*(

*x*,

_{p}*y*,

_{p}*z*), in the focal region of a diffraction-limited, high numerical aperture lens with an arbitrary polarized uniform incident illumination as shown in Fig. 1, is given by the Debye-Wolf integral

_{p}*k*=2

*πn*/

*λ*is the wavenumber of the illumination,

**a**(

*s*,

_{x}*s*) is the strength vector of a ray at the Gaussian reference sphere converging to the focal point,

_{y}*s*,

_{x}*s*and

_{y}*s*are the Cartesian components of the unit vector

_{z}**s**along this ray and Θ is the solid angle associated with all the rays which reach the image space through the exit pupil of the lens [1–3].

*r*,

*θ*,

*ϕ*(

*r*>0, 0≤

*θ*<

*π*, 0≤

*ϕ*<2

*π*) with the polar axis

*θ*=0 in the

*z*direction and with the azimuth

*ϕ*=0 containing the linearly polarized incident vector are introduced. The unit vector

**s**along a ray and the point

*p*(

*x*,

_{p}*y*,

_{p}*z*) in the focal region can be expressed as

_{p}*A*is a constant,

*α*=

*θ*

_{max}and

*ρ*=

_{p}*r*sin

_{p}*θ*,

_{p}*z*=

_{p}*r*cos

_{p}*θ*. We note that the dummy variables of the double integrals in Eq. (4) are given in spherical coordinates while the field distributions are given in cylindrical coordinates.

_{p}## 3. Eigenfunction representation of the fields of the Debye-Wolf focusing integral

*E*in Eq. (4a). Afterwards we show the final result which we can obtain in a similar manner for

_{x}*E*in Eq. (4b) and

_{y}*E*in Eq. (4c). We start by noting that the defocus term, exp[

_{z}*ikz*cos

_{p}*θ*], in Eq. (4) could be written as [14]

*J*(·) are Bessel functions of the first kind of order

_{m}*m*. Substituting Eq. (5) in Eq. (4a), we obtain

*u*=sin

*θ*which results in

*dθ*=

*du*/cos

*θ*.

*α*′=sin

*α*. Since exp(

*iθ*)=cos

*θ*+

*i*sin

*θ*, cos

*mθ*=

*T*(cos

_{m}*θ*) and sin[(

*m*+1)

*θ*]=sin

*θU*(cos

_{m}*θ*), where

*T*(·) and

_{m}*U*(·) are Chebyshev polynomials of the first and second types respectively [15], we can write

_{m}*a*(

^{x}_{m}*u*,

*ϕ*) is space-limited, so we could expand it in terms of generalized circular prolate spheroidal functions, which are eigenfunctions of the two-dimensional finite Hankel transform [16, 17]

*c*is a parameter equal to or larger than the radial space-bandwidth product of

*a*(

^{x}_{m}*u*,

*ϕ*) for all values of 0<

*ϕ*≤2

*π*[18].

*ikρ*cos

_{p}u*ϕ*]=∑

^{∞}

_{q}=-∞

*i*(

^{q}J_{q}*kρ*)exp(

_{p}u*iqϕ*) [18] and the fact that cos(

*ϕ*-

*ϕ*)=cos(

_{p}*ϕ*) and ∫

_{p}-ϕ^{2π}

_{0}exp(

*i*(

*N*-

*q*)

*ϕ*)

*dϕ*=

*δ*where

_{Nq}*δ*is the Kronecker delta function, we obtain

_{Nq}*N*<0 we could write

*J*(

_{N}*x*)=(-1)

^{N}*J*

_{|N|}(

*x*) and

*i*(-1)

^{N}*=*

^{N}*i*=

^{-N}*i*to obtain

^{|N|}*ω*

_{max}[18], and by comparing Eq. (14) and Eq. (15) and setting

*r*

_{0}=

*α*′,

*ω*=

*kρ*and Ω=

_{p}*kρ*

_{pmax}, we obtain

*A*and

^{y}_{m,N,n}*A*are the expansion coefficients of

^{z}_{m,N,n}*it constitutes the main result of this paper*.

## 4. Arbitrary polarization

*y*and

*z*counterparts. The general form for the strength vector takes the form

*g*(

*u*,

*ϕ*) and Ψ(

*u*,

*ϕ*) describe the amplitude and phase variation introduced to the incident field distribution

**A**is the matrix given by [19

19. P. Török, P.D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. **148**300–315 (1998) [CrossRef]

23. P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express **12**, 3605–3617 (2004). [CrossRef] [PubMed]

## 5. Physical properties of the eigenfunction expansion

*J*(

_{m}*kz*), exp(

_{p}*iNϕ*), Φ

_{p}_{|N|,n}(

*α*′

*ρ*/

_{p}*ρ*

_{pmax}) are separable in cylindrical coordinates which could simplify analysis involving fields in the focal region of a high NA focusing system.

_{|N|,0}(

*r*,

*c*) maximize the fractional energy within a circular region of radius

*r*

_{0}over the class of all bandlimited functions [18]. Thus our summation over

*N*has maximum energy packing properties in both radial and azimuthal directions when

*n*=0. More generally the eigenvalues,

*λ*

_{|N|,n}, describe the fractional energy within the circular region for each generalized prolate spheroidal function.

*N*|≥|

*N*

_{0}|,

*n*≥

*n*

_{0}. We also note, as shown in Fig. 2(b), that Bessel functions of the same argument, but of increasing orders, decrease to very small values, say 10

^{-4}, after some order

*m*≥

*m*

_{0}. Therefore our expansion demonstrates fast convergence in the azimuthal, radial and axial directions.

*z*=0) eigenfunctions which reduce to the generalized prolate spheroidal functions, whilst defocused eigenfunctions are further modulated by a Bessel function dependent on the axial coordinate. For low numerical aperture systems the polarization properties of light become less important often allowing a scalar treatment to be used. Under such circumstances we can interpret the field distributions shown in Fig. 3 as the true eigenfunctions of the focusing operation. At higher numerical apertures the distributions shown are not strictly eigenfunctions of the Debye-Wolf integral since they are scalar functions, however they do remain eigenfunctions on a component-wise basis i.e. if

_{p}*x*-polarized light with amplitude distribution given by the (

*N*,

*n*)

^{th}order were focused, the

*x*component of the output field would also have the same (although scaled) distribution.

*N*=0 modes have a central focal spot. Put another way the higher order functions contain higher spatial frequencies however energy is not concentrated as efficiently into the central region. Consequently more complicated masking optics will, in general, require more terms to be calculated in the eigenfunction expansion to accurately determine the field in the focal region.

*λ*

_{|N|,n}, with the numerical aperture of the focusing system. It can be seen that for a given order (

*N*,

*n*) the eigenvalue decreases as the NA decreases. This dependence means that higher orders are energetically less significant in the focused distribution and hence the dominant orders are those with lower spatial frequencies or equivalently are less tightly packed. The dominant modes in the focal region hence provide a further means to determine the resolution of the optical system.

## 6. Numerical examples

21. P. E. Falloon, *Hybrid Computation of the Spheroidal Harmonics and Application to the Generalized Hydrogen Molecular Ion Problem* (University of Western Australia, 2001). [PubMed]

22. W. Latham and M. Tilton, “Calculation of prolate functions for optical analysis,” Appl. Opt. **26**, 2653–2658 (1987). [CrossRef] [PubMed]

*c*and suitable finite limits for the three summations in Eq. (16). The parameter

*c*has to be equal to or larger than the radial space-bandwidth product of the functions

*a*

^{(·)}

_{m}(

*u*,

*ϕ*) in Eq. (9) and Eq. (17) for all values 0<

*ϕ*≤2

*π*. In addition,

*c*=

*α*′

*kρ*

_{pmax}, so for a given value of the NA,

*c*determines the radial field of view,

*ρ*

_{pmax}, in the plane of interest. For the following numerical examples, we found that

*c*=20 satisfied these two requirements.

*c*=20, |

*N*

_{0}|=23 and

*n*

_{0}=8 are appropriate limits for the summations with respect to

*N*and

*n*, respectively. From Fig. 2(b), we note that for a defocus distance

*z*=

*λ*,

*m*

_{0}=14 is an appropriate limit for the summation with respect to

*m*.

*z*=

*λ*, respectively, obtained by evaluating our eigenfunction expansion, Eq. (16), with finite summation limits and by direct integration. The actual calculated optical distribution is also shown in the insets.

*ε*given by

*I*

^{direct}

_{k,l}and

*I*

^{expansion}

_{k,l}are the intensities at point (

*k*,

*l*) obtained by evaluating the Debye-Wolf integral and our eigenfunction expansion, respectively, and (

*K*,

*L*) are the maximum number of points in the plane of interest. On applying Eq. (21) to the above numerical examples, we found

*ε*equal to 0.0253% and 0.065% at the Gaussian focal plane (Fig. 5) and at a defocus distance

*z*=

*λ*(Fig. 6), respectively. This overall error is very small, bearing in mind the relatively small number of terms used to evaluate Eq. (16). Furthermore it is found the variation of the total percentage error as the NA is increased from 0 to 0.966 is of order 0.003% and is hence negligible.

## 7. Conclusions

## Acknowledgments

*Mathematica*code to compute Slepian’s Generalized Spheroidal functions. S. Sherif thanks Costel Flueraru, Shoude Chang and Linda Mao for their support through the Optics Group at the Institute for Microstructural Sciences, National Research Council Canada.

## References and links

1. | B. Richards “Diffraction in systems of high relative aperture,” in |

2. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London A , |

3. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London A , |

4. | G. N. Watson, |

5. | I. S. Gradshtyen and I. M. Ryzhik, |

6. | G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. |

7. | R. Kant, “An analytical solution of vector diffraction for focusing optical systems with seidel aberrations” J. Mod. Opt. |

8. | P. Török, S.J. Hewlett, and P. Varga, “On the series expansion of high aperture, vectorial diffraction integrals,” J. Mod. Opt. |

9. | S. S. Sherif and P. Török, “Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula,” J. Mod. Opt. |

10. | J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A |

11. | C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. |

12. | S. S. Sherif and P. Török, “Pupil plane masks for super-resolution in high-numerical-aperture focusing,” J. Mod. Opt. |

13. | M. R. Foreman, Department of Physics, Imperial College London, Prince Consort Rd. London SW7 2BZ, United Kingdom, S. S. Sherif, P.R.T. Munro, and P. Török are preparing a manuscript to be called “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region” |

14. | G. Arfken, |

15. | M. Abramowitz and I. Stegun, |

16. | D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Sys. Tech. J. |

17. | J. C. Heurtley, “Hyperspheroidal functions-optical resonators with circular mirrors,” in |

18. | B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on the prolate functions,” |

19. | P. Török, P.D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. |

20. | S. Zhang and J. Jin, |

21. | P. E. Falloon, |

22. | W. Latham and M. Tilton, “Calculation of prolate functions for optical analysis,” Appl. Opt. |

23. | P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1960) Diffraction and gratings : Diffraction theory

(180.0180) Microscopy : Microscopy

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: February 21, 2008

Manuscript Accepted: February 26, 2008

Published: February 28, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Sherif S. Sherif, Matthew R. Foreman, and Peter Török, "Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system," Opt. Express **16**, 3397-3407 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-5-3397

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### References

- B. Richards "Diffraction in systems of high relative aperture," in Astronomical Optics and Related Subjects, Z. Kopal, ed., (North Holland Publishing Company, 1955), pp. 352-359.
- E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A, 253, 349-357 (1959). [CrossRef]
- B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London A, 253, 358-379 (1959). [CrossRef]
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed. (Cambridge University Press, 1952). [PubMed]
- I. S. Gradshtyen and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 1980).
- G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979) [CrossRef]
- R. Kant, "An analytical solution of vector diffraction for focusing optical systems with seidel aberrations" J. Mod. Opt. 40, 2293-2310 (1993). [CrossRef]
- P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997). [CrossRef]
- S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005). [CrossRef]
- J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, "Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system," J. Opt. Soc. Am. A 20, 2281-2292 (2003). [CrossRef]
- C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997). [CrossRef]
- S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).
- M. R. Foreman, Department of Physics, Imperial College London, Prince Consort Rd. London SW7 2BZ, United Kingdom, S. S. Sherif, P. R. T. Munro and P. Török are preparing a manuscript to be called "Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region"
- G. Arfken, Mathematical Methods for Physicists, 3rd Edition (Academic Press, 1985).
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).
- D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).
- J. C. Heurtley, "Hyperspheroidal functions-optical resonators with circular mirrors," in Proceedings of Symposium on Quasi-Optics, J. Fox, ed., (Polytechnic Press, 1964), pp. 367-371.
- B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on the prolate functions," Progress in Optics 9, E. Wolf, ed., (Pergamon, 1971), pp. 311- 407.
- P. Török, P. D. Higdon and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148,300-315 (1998) [CrossRef]
- S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1996).
- P. E. Falloon, Hybrid Computation of the Spheroidal Harmonics and Application to the Generalized Hydrogen Molecular Ion Problem (University of Western Australia, 2001). [PubMed]
- W. Latham and M. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653-2658 (1987). [CrossRef] [PubMed]
- P. Török and P. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy," Opt. Express 12, 3605-3617 (2004).http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3605 [CrossRef] [PubMed]

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